Concentration phenomena of solutions for critical perturbed Hénon problems

Abstract

The main aim in this paper is to carry out a comprehensive research on the critical Hénon problem$\{\begin{array}{cc}-\Delta u=|x{|}^{\alpha }{u}^{p_{\alpha }}+ϵk(x)u^q& \text{in}\Omega ,\\ u>0& \text{in}\Omega ,\\ u=0& \text{on}\partial \Omega ,\end{array}$ where $\alpha >0$, $p_{\alpha }=\frac{N+2+2\alpha }{N-2}$, $q\ge 1$, $ϵ>0$, $k\left(x\right)\in C^2\left(\overline{\Omega }\right)$, Ω is a smooth bounded domain containing the origin in $R^N$, $N\ge 3$. Based on Lyapunov-Schmidt reduction argument, we provide some sufficient conditions for the existence of concentrating solutions without any condition on the Robin function. The main results depend on the non-resonant case that $k\left(0\right)\ne 0\text{and}q\ne \frac{N+2}{N-2}$ and the resonant case that $k\left(0\right)=0\text{or}q=\frac{N+2}{N-2}$. The novelty in our study is significantly different from the case that $\alpha =0$.